The Promotion Of Banach-Mackey Property And (LC) Property On The Direct Sum Spaces

Topological linear spaces theory plays very important role in functional analysis theory, which has been widely applicated in Banach spaces geometric theory and dual theory. Up to now, the research on topological linear spaces has been perfect, but the studying on direct sum spaces is no very perfect. Basing on paper[1][2][6], we defined some kinds topologies on the direct sum spaces, then we studied the polar theory and dual theory on the direct sum spaces under those kinds topologies which we defined in this paper, furthermore we generalized the Banach-Mackey property and(LC) property to the direct sum spaces.Chapter 1: The knowledge of preparation.Chapter 2: Through constructing some kind topologies in direct sum space, we discuss the bipolar theorem, Alaoglu-Boubingrbaki theorem and Mackey-Arens theorem on the direct sum spaces, then we generalized the Banach-Mackey property in direct sum space. From this we get several results.Chapter 3: We discuss the conception of Banach disk,(L) property and(LC)property on the direct sum spaces, then by Proofing the lemma 3.2.1 we generalized some equal statements of paper[9] to the direct sum spaces.Chapter 4: We give a condition of convex compactness property for sequentially complete spaces and study the convex compactness property in the natural pair dual spaces.